Subgroups — C₂₃×#8618 #19156244

Order 5888.

454 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1800	C₂	1	2	yes	maximal
3	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1292	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₂₃	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#436	—	2	1	no	—
9	C₂₃×#436	—	2	1	no	—
10	C₂₃×#436	—	2	1	no	—
11	C₂₃×#436	—	2	1	no	—
12	C₂₃×#423	C₂²	1	2	yes	—
13	C₂₃×#423	C₂²	1	2	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#100	—	2	2	no	—
16	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#343	—	2	2	no	—
19	C₂₃×#335	—	2	1	no	—
20	C₂₃×#335	—	2	1	no	—
21	C₂₃×#334	—	2	1	no	—
22	C₂₃×#334	—	2	1	no	—
23	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	4	4	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₄×D₄×C₂₃	—	2	1	no	—
34	C₄×D₄×C₂₃	—	2	1	no	—
35	C₄×D₄×C₂₃	—	2	1	no	—
36	C₄×D₄×C₂₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	—	2	1	no	—
41	C₂₃×C₄⋊C₈	—	2	1	no	—
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	—	2	2	no	—
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×#100	—	4	4	no	—
47	C₂₃×C₂²⋊C₈	—	2	2	no	—
48	C₂₃×C₂²⋊C₈	—	2	2	no	—
49	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₂⁴×C₂₃	C₄⋊C₄	1	2	yes	—
54	C₂×D₄×C₂₃	—	2	1	no	—
55	C₂×D₄×C₂₃	—	2	1	no	—
56	C₂×D₄×C₂₃	—	2	2	no	—
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	—	2	2	no	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	—	4	2	no	—
62	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
66	M₁₆×C₂₃	—	4	4	no	—
67	M₁₆×C₂₃	—	4	4	no	—
68	M₁₆×C₂₃	—	4	2	no	—
69	M₁₆×C₂₃	—	4	2	no	—
70	M₁₆×C₂₃	—	4	2	no	—
71	M₁₆×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	4	2	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂²⋊C₄×C₂₃	—	2	2	no	—
85	C₂²⋊C₄×C₂₃	—	2	2	no	—
86	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₂₃	#41	1	1	yes	characteristic
89	C₄²×C₂₃	#41	1	1	yes	characteristic
90	#8618	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	—	2	4	no	—
93	C₂³×C₂₃	—	4	2	no	—
94	C₂³×C₂₃	—	4	2	no	—
95	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	D₄×C₂₃	—	4	2	no	—
97	D₄×C₂₃	—	4	1	no	—
98	D₄×C₂₃	—	4	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	4	1	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₂₃	#137	1	2	yes	—
115	C₈×C₂₃	—	2	1	no	—
116	C₈×C₂₃	—	2	1	no	—
117	C₈×C₂₃	—	2	1	no	—
118	C₈×C₂₃	—	4	2	no	—
119	C₈×C₂₃	—	8	2	no	—
120	C₈×C₂₃	—	8	2	no	—
121	C₈×C₂₃	#100	1	2	yes	complemented
122	C₂×#436	C₂×C₂₃	1	1	yes	characteristic
123	#1800	C₂×C₂₃	1	2	yes	—
124	#1567	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#1566	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#1292	C₂×C₂₃	1	2	yes	complemented
127	C₂²×C₂₃	—	2	1	no	—
128	C₂²×C₂₃	—	2	1	no	—
129	C₂²×C₂₃	—	4	2	no	—
130	C₂²×C₂₃	—	4	2	no	—
131	C₂²×C₂₃	—	4	2	no	—
132	C₂²×C₂₃	—	8	4	no	—
133	C₂²×C₂₃	#380	1	1	yes	characteristic
134	C₂²×C₂₃	#428	1	2	yes	—
135	C₄×C₂₃	—	2	1	no	—
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	4	2	no	—
139	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
140	C₄×C₂₃	#413	1	1	yes	characteristic
141	C₄×C₂₃	#468	1	2	yes	—
142	C₄×C₂₃	#469	1	1	yes	characteristic
143	C₄×C₂₃	#470	1	1	yes	characteristic
144	C₂²×M₁₆	C₄×C₂₃	1	2	yes	—
145	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
146	#436	—	2	1	no	—
147	#436	—	2	1	no	—
148	#436	—	2	1	no	—
149	#436	—	2	1	no	—
150	#423	C₂²×C₂₃	1	2	yes	—
151	#423	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×#100	—	2	2	no	—
154	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
156	#343	—	2	2	no	—
157	#335	—	2	1	no	—
158	#335	—	2	1	no	—
159	#334	—	2	1	no	—
160	#334	—	2	1	no	—
161	C₂×C₂₃	—	2	2	no	—
162	C₂×C₂₃	—	8	2	no	—
163	C₂×C₂₃	#1800	1	1	yes	characteristic
164	C₂×C₂₃	#1809	1	1	yes	characteristic
165	C₂×C₂₃	#1813	1	1	yes	characteristic
166	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
167	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	4	4	no	—
173	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
175	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	Q₈×C₂₃	1	2	yes	—
182	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	#100	—	4	4	no	—
190	C₂²⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
193	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
194	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
195	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
196	C₂₃	#8618	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	C₄⋊C₄×C₂₃	1	2	yes	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
207	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
210	M₁₆	—	4	4	no	—
211	M₁₆	—	4	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
231	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₄²	#41×C₂₃	1	1	yes	characteristic
233	C₄²	#41×C₂₃	1	1	yes	characteristic
234	C₂³	—	2	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	D₄	—	4	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
252	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
253	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
254	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
257	C₂×C₄	C₂₃×#137	1	2	yes	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	4	2	no	—
262	C₈	—	8	2	no	—
263	C₈	—	8	2	no	—
264	C₈	C₂₃×#100	1	2	yes	complemented
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	8	4	no	—
271	C₂²	C₂₃×#380	1	1	yes	characteristic
272	C₂²	C₂₃×#428	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
278	C₄	C₂₃×#413	1	1	yes	characteristic
279	C₄	C₂₃×#468	1	2	yes	—
280	C₄	C₂₃×#469	1	1	yes	characteristic
281	C₄	C₂₃×#470	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	8	2	no	—
284	C₂	C₂₃×#1800	1	1	yes	characteristic
285	C₂	C₂₃×#1809	1	1	yes	characteristic
286	C₂	C₂₃×#1813	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂₃×#1567	C₂	semidirect
4. C₂₃×#1566	C₂	semidirect
5. C₂₃×#1292	C₂	semidirect
90. #8618	C₂₃	direct (N × H)
121. C₈×C₂₃	#100	semidirect
124. #1567	C₂×C₂₃	semidirect
125. #1566	C₂×C₂₃	semidirect
126. #1292	C₂×C₂₃	semidirect
196. C₂₃	#8618	direct (N × H)
264. C₈	C₂₃×#100	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#436	2
C₂₃×#1800	2
C₂₃×#1567	2
C₂₃×#1566	2
C₂₃×#1292	2
#8618	23

Subgroups — C23×#8618 #19156244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8618 #19156244